Culver, Dominic Leon; Kong, Hana Jia; Quigley, J. D.

Algebraic slice spectral sequences

Doc. Math. 26, 1085-1119 (2021)
DOI: 10.25537/dm.2021v26.1085-1119
Communicated by Mike Hill


For certain motivic spectra, we construct a square of spectral sequences relating the effective slice spectral sequence and the motivic Adams spectral sequence. We show the square can be constructed for connective algebraic K-theory, motivic Morava K-theory, and truncated motivic Brown-Peterson spectra. In these cases, we show that the \(\mathbb{R}\)-motivic effective slice spectral sequence is completely determined by the \(\rho\)-Bockstein spectral sequence. Using results of Heard, we also obtain applications to the Hill-Hopkins-Ravenel slice spectral sequences for connective Real K-theory, Real Morava K-theory, and truncated Real Brown-Peterson spectra.

Mathematics Subject Classification

14F42, 55P42, 55P91, 55T05, 55T15


motivic Adams spectral sequence, slice spectral sequence, algebraic slice spectral sequence


  • 1. A. Ananyevskiy, O. Röndigs, and P. A. Østvær. On very effective Hermitian \(K\)-theory. Math. Z. 294(3-4):1021-1034, 2020, DOI 10.1007/s00209-019-02302-z, zbl 1453.14064, MR4074031, arxiv 1712.01349.
  • 2. M. F. Atiyah. K-theory and reality. Quart. J. Math. Oxford Ser. (2) 17(1):367-386, 1966, DOI 10.1093/qmath/17.1.367, zbl 0146.19101, MR0206940.
  • 3. T. Bachmann. The generalized slices of Hermitian K-theory. J. Topol. 10(4):1124-1144, 2017, DOI 10.1112/topo.12032, zbl 1453.14065, MR3743071, arxiv 1610.01346.
  • 4. M. Behrens and J. Shah. \(C_2\)-equivariant stable homotopy from real motivic stable homotopy. Ann. K-Theory 5(3):411-464, 2020, DOI 10.2140/akt.2020.5.411, zbl 07237238, MR4132743, arxiv 1908.08378.
  • 5. E. Belmont, B. Guillou, and D. Isaksen. \(C_2\)-equivariant and R-motivic stable stems, II. Proc. Amer. Math. Soc. 149(1):53-61, 2021, DOI 10.1090/proc/15167, zbl 07301316, MR4172585, arxiv 2001.02251.
  • 6. E. Belmont and D. C. Isaksen. \(R\)-motivic stable stems, 2020, arxiv 2001.03606.
  • 7. J. M. Boardman. Conditionally convergent spectral sequences. In Homotopy invariant algebraic structures (Baltimore, MD, 1998), volume 239 of Contemp. Math., pages 49-84. Amer. Math. Soc., Providence, RI, 1999, zbl 0947.55020, MR1718076.
  • 8. S. Borghesi. Algebraic Morava K-theories. Invent. Math. 151(2):381-413, 2003, DOI 10.1007/s00222-002-0257-4, zbl 1030.55003, MR1953263.
  • 9. S. Borghesi. Algebraic Morava \(K\)-theory spectra over perfect fields. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 8(2):369-390, 2009, DOI 10.2422/2036-2145.2009.2.06, zbl 1179.14019, MR2548251.
  • 10. D. Dugger. An Atiyah-Hirzebruch spectral sequence for \(KR\)-theory. \(K\)-Theory 35(3-4):213-256, 2005, DOI 10.1007/s10977-005-1552-9, zbl 1109.14024, MR2240234, arxiv math/0304099.
  • 11. D. Dugger and D. C. Isaksen. Motivic cell structures. Algebr. Geom. Topol. 5(2):615-652, 2005, DOI 10.2140/agt.2005.5.615, zbl 1086.55013, MR2153114, arxiv math/0310190.
  • 12. B. J. Guillou, M. A. Hill, D. C. Isaksen, and D. C. Ravenel. The cohomology of \(C_2\)-equivariant \(A(1)\) and the homotopy of \(ko_{C_2}\). Tunis. J. Math. 2(3):567-632, 2019, DOI 10.2140/tunis.2020.2.567, zbl 1440.14124, MR4041284, arxiv 1708.09568.
  • 13. J. J. Gutiérrez, O. Röndigs, M. Spitzweck, and P. A. Østvær. Motivic slices and coloured operads. J. Topol. 5(3):727-755, 2012, DOI 10.1112/jtopol/jts015, zbl 1258.18012, MR2971612, arxiv 1012.3301.
  • 14. D. Heard. On equivariant and motivic slices. Algebr. Geom. Topol. 19(7):3641-3681, 2019, DOI 10.2140/agt.2019.19.3641, zbl 1441.14077, MR4045363, arxiv 1807.09092.
  • 15. M. A. Hill. Ext and the motivic Steenrod algebra over \(\mathbb{R} \). J. Pure Appl. Algebra 215(5):715-727, 2011, DOI 10.1016/j.jpaa.2010.06.017, zbl 1222.55014, MR2747214, arxiv 0904.1998.
  • 16. M. A. Hill, M. J. Hopkins, and D. C. Ravenel. On the nonexistence of elements of Kervaire invariant one. Ann. of Math. (2) 184(1):1-262, 2016, DOI 10.4007/annals.2016.184.1.1, zbl 1366.55007, MR3505179.
  • 17. M. Hoyois. From algebraic cobordism to motivic cohomology. J. Reine Angew. Math. 702:173-226, 2015, DOI 10.1515/crelle-2013-0038, zbl 1382.14006, MR3341470, arxiv 1210.7182.
  • 18. M. Hoyois, S. Kelly, and P. A. Østvær. The motivic Steenrod algebra in positive characteristic. J. Eur. Math. Soc. (JEMS) 19(12):3813-3849, 2017, DOI 10.4171/JEMS/754, zbl 1386.14087, MR3730515, arxiv 1305.5690.
  • 19. P. Hu and I. Kriz. Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence. Topology 40(2):317-399, 2001, DOI 10.1016/S0040-9383(99)00065-8, zbl 0967.55010, MR1808224.
  • 20. P. Hu, I. Kriz, and K. Ormsby. Convergence of the motivic Adams spectral sequence. J. K-Theory 7(3):573-596, 2011, DOI 10.1017/is011003012jkt150, zbl 1309.14018, MR2811716.
  • 21. D. C. Isaksen and P. A. Østvær. Motivic stable homotopy groups. In Handbook of homotopy theory, CRC Press/Chapman Hall Handb. Math. Ser., pages 757-791. CRC Press, Boca Raton, FL [2020], DOI 10.1201/9781351251624-18, zbl 07303324, MR4197998.
  • 22. D. C. Isaksen, G. Wang, and Z. Xu. More stable stems, 2020, arxiv 2001.04511.
  • 23. H. J. Kong. The \(C_2\)-effective spectral sequence for \(C_2\)-equivariant connective real K-theory, 2020, arxiv 2004.00806.
  • 24. J. I. Kylling. Hermitian K-theory of finite fields via the motivic Adams spectral sequence. Master's thesis, University of Oslo, Norway, 2015.
  • 25. J. I. Kylling. Recursive formulas for the motivic Milnor basis. New York J. Math. 23(250399):49-58, 2017, zbl 1361.55023, MR3611073, arxiv 1704.00031.
  • 26. J. I. Kylling and G. M. Wilson. Strong convergence in the motivic Adams spectral sequence, 2019, arxiv 1901.03399.
  • 27. M. Levine. The homotopy coniveau tower. J. Topol. 1(1):217-267, 2008, DOI 10.1112/jtopol/jtm004, zbl 1154.14005, MR2365658, arxiv math/0510334.
  • 28. M. Levine and G. S. Tripathi. Quotients of MGL, their slices and their geometric parts. Doc. Math., Extra Vol. Alexander S. Merkurjev's sixtieth birthday, 407-442, 2015,, zbl 1366.14021, MR3404387.
  • 29. L. Mantovani. Localizations and completions in motivic homotopy theory, 2018, arxiv 1810.04134.
  • 30. C. May. A structure theorem for \(RO(C_2)\)-graded Bredon cohomology. Algebr. Geom. Topol. 20(4):1691-1728, 2020, DOI 10.2140/agt.2020.20.1691, zbl 1447.55009, MR4127082, arxiv 1804.03691.
  • 31. F. Morel. Suite spectrale d'Adams et invariants cohomologiques des formes quadratiques. C. R. Acad. Sci. Paris Sér. I Math. 328(11):963-968, 1999, DOI 10.1016/S0764-4442(99)80306-1, zbl 0937.19002, MR1696188.
  • 32. F. Morel and V. Voevodsky. \(A^1\)-homotopy theory of schemes. Inst. Hautes Études Sci. Publ. Math. 90:45-143, 1999, DOI 10.1007/BF02698831, zbl 0983.14007, MR1813224.
  • 33. K. M. Ormsby. Motivic invariants of \(p\)-adic fields. J. K-Theory 7(3):597-618, 2011, DOI 10.1017/is011004017jkt153, zbl 1258.14025, MR2811717, arxiv 1002.5007.
  • 34. K. M. Ormsby and P. A. Østvær. Motivic Brown-Peterson invariants of the rationals. Geom. Topol. 17(3):1671-1706, 2013, DOI 10.2140/gt.2013.17.1671, zbl 1276.55023, MR3073932, arxiv 1208.5007.
  • 35. O. Röndigs and P. A. Østvær. Slices of Hermitian K-theory and Milnor's conjecture on quadratic forms. Geom. Topol. 20(2):1157-1212, 2016, DOI 10.2140/gt.2016.20.1157, zbl 1416.19001, MR3493102, arxiv 1311.5833.
  • 36. O. Röndigs, M. Spitzweck, and P. A. Østvær. The first stable homotopy groups of motivic spheres. Ann. of Math. (2) 189(1):1-74, 2019, DOI 10.4007/annals.2019.189.1.1, zbl 1406.14018, MR3898173, arxiv 1604.00365.
  • 37. M. Spitzweck. Relations between slices and quotients of the algebraic cobordism spectrum. Homology Homotopy Appl. 12(2):335-351, 2010, DOI 10.4310/HHA.2010.v12.n2.a11, zbl 1209.14019, MR2771593, arxiv 0812.0749.
  • 38. M. Spitzweck and P. A. Østvær. Motivic twisted K-theory. Algebr. Geom. Topol. 12(1):565-599, 2012, DOI 10.2140/agt.2012.12.565, zbl 1282.14040, MR2916287, arxiv 1008.4915.
  • 39. S.-T. Stahn. The motivic Adams-Novikov spectral sequence at odd primes over \(\mathbb{C}\) and \(\mathbb{R} . 2016\), arxiv 1606.06085.
  • 40. V. Voevodsky. A1-homotopy theory. In Proceedings of the International Congress of Mathematicians, Doc. Math., Extra Vol. ICM Berlin 1998, vol. I, pages 579-604. Berlin, 1998,, zbl 0907.19002, MR1648048.
  • 41. V. Voevodsky. Open problems in the motivic stable homotopy theory. I. Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., 3, I, pages 3-34, 1998, zbl 1047.14012, MR1977582.
  • 42. V. Voevodsky. Reduced power operations in motivic cohomology. Publ. Math. Inst. Hautes Études Sci. 98(1):1-57, 2003, DOI 10.1007/s10240-003-0009-z, zbl 1057.14027, MR2031198, arxiv math/0107109.
  • 43. G. M. Wilson and P. A. Østvær. Two-complete stable motivic stems over finite fields. Algebr. Geom. Topol. 17(2):1059-1104, 2017, DOI 10.2140/agt.2017.17.1059, zbl 1361.14020, MR3623682, arxiv 1601.06398.
  • 44. N. Yagita. Applications of Atiyah-Hirzebruch spectral sequences for motivic cobordism. Proc. London Math. Soc. (3) 90(3):783-816, 2005, DOI 10.1112/S0024611504015084, zbl 1086.55005, MR2137831.


Culver, Dominic Leon
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Kong, Hana Jia
Department of Mathematics, 5734 S University Ave, Chicago IL, 60637, USA
Quigley, J. D.
Department of Mathematics, Cornell University Ithaca, NY, U.S.A.